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As we move into another week of “school at a distance”, I’d like to offer you a game for intermediate students. This is a game that requires a partner and a regular 6-sided die (although a 10-sided one will make things more interesting!).

Full instructions for play are on the Roll The Bigger Productgame board, but the goal is to take turns placing numbers in each of the positions in the 2-digit factors to create the largest possible product. You get to discard 2 rolls — throwing them into the trashcan — to be even more strategic! When all 6 positions are filled, calculate the product and compare it to your partner’s. The larger product wins.

To add complexity to the game, try placing decimals between both double digit factors — or harder still, within just one of the 2 factors.

Players start on opposite sides of the hexagonal game board and roll a regular 6-sided die, then choose whether to add the number shown or to subtract it, moving only to touching spaces.

The goal of the game is to land on the opposing player’s start space.

As the days become warmer and drier here I am imagining a version of this game played outside on an interlocking brick driveway or a drawn with … you guessed it… sidewalk chalk. :o) But perhaps that’s a bit ambitious.

Regardless of how you choose to play, students working on mastering addition and subtraction will enjoy Cross Over, and the combination of strategy and luck will ensure that even older children and parents will find the game accessible and fun.

Today marks the beginning of a new kind of school — the stay-at-home kind. More than ever we are going to need to be flexible and patient and kind to our kids, their parents and our teacher colleagues. We are in uncharted waters… but not without good will!

I wanted to offer up a couple of simple dice games for you to play at home, to build number sense and computational fluency while having fun…

Reach for the Top You can play this game alone, with a partner or against a partner. Print a copy of the Reach for the Top gridavailable here.

How to play: Roll two 6-sided dice. Find the sum. Write the equation in the box over the sum (or colour it in). The game is over when one of the sums reaches the top! (See the sample game below.)

Order of Operations Bowling: You can play this game alone, with a partner or against a partner. You need one 6-sided die and the optional Bowling Pinrecording sheet attached. The object of the game is to “knock down” all the bowling pins from 1 to 10.

How to play: Roll the die 3 times. Record the 3 numbers. Use these three numbers — in any order — to create an equation with an answer of 1. You must use all three numbers. Once you’ve found an equation (or 2 or 3!) with an answer of 1, cross off the bowling pin with the number 1 on it. Now move on to the number 2, then 3, then 4… until you have created equations with all the answers from 1 to 10. Each time you find an equation, you can knock down the pin with that number on it.

If you can knock down all 10 bowling pins with one set of numbers, you get a “strike”. If not, roll the dice 3 more times to get a new set of numbers and continue. Two sets of numbers earns you a “spare”. How many different operations can you use?

In the sample round below, Player A rolled a 1, a 5 and a 6. She used all 3 numbers to create equations (a whole bunch of them!) with an answer of 1. She chooses one of the equations and knocks down the 1 pin. She then moves on to create equations with an answer of 2.

If you’re outside (or wish you were!) here’s another math game worth playing. It’s called Testing the Limits — and I’ve adapted it from BEAM Maths of the Month.

This version is adaptable for all grade levels and can be played outside with sidewalk chalk (or indoors on paper!).

To play, you need sidewalk chalk and a die (6-sided is fine, 10-sided is better!). You can play this game with a partner or alone. Here’s how…

ROUND 1: Roll the die 3 times and make a 3-digit number. Roll 3 more times and make a second 3 digit number. Put both numbers on the same number line. These are your “limits”.

ROUND 2: Now roll 6 more times. Make 2 different 3-digit numbers that fit within the limits from ROUND 1. Plot them on the same number line as ROUND 1. If you can do it, these new numbers become your new limits and you can move onto ROUND 3. If not… the game is over! Check out my sample game below.

Try using just two 2-digit numbers for younger students, or even decimal numbers for older students. Consider trying this with negative numbers, or even one negative and one positive to explore both sides of zero.

For those of you who are looking for ways to play together and build mathematical thinking and skills at the same time, consider this simple hopscotch game. You’ll need sidewalk chalk and a small stone. Children can play alone or in partners.

Go outside. Draw a large square on the pavement. Divide the square into at least 9 smaller squares, as shown below. This is called a matrix.

In each of the smaller squares record a number from 1 to 9. You can put them in any order. Now take a small stone and toss it onto the matrix. This is your starting square. From here, you must jump to the number that adds to give you 10.

In the game below, a child has thrown a stone onto the number 8. She stands on the number 8 then and has to jump to get to the number 2 — the missing part to get to 10.

If there’s another child nearby, they should record the equation that matches the jump. (8 + 2 = 10)

Player 2 (if there is one) takes his turn, throwing a stone and jumping from that number to the missing part to make ten. Player A records the equation.

The first player to hop on all the combinations is the winner. (And yes, landing on a 5 gets you a double jump!)

If you’re stuck inside, make the matrix on a sheet of paper and toss coins — or even Cheerios! — instead of jumping from number to number. Toss the first coin, say the number you’ve landed on, then say what the missing part is to get to the desired sum.

Of course you can change the numbers to suit the age and stage of the players…

Consider a double-digit version (Get to 100!) or even a decimal version (Get to 5.0!). The sky’s the limit. I’ve included line masters for each of these games — and a blank grid, too — for you to use as inspiration.

We have always been partners in the mathematics education of our children — and now we find ourselves in a time that demands even more collaboration between home and school. Ensuring that our children have meaningful math learning experiences when we are inside and inundated with technological distractions can be difficult.

To that end I am inviting you to explore the following resource for parents of primary aged children. It was written years ago in a partnership between the BC Ministry of Education and a group of respected BC educators to support parents of young children to find and highlight mathematics in their daily lives. From sorting and counting to estimating and measuring, Math For Families has dozens of simple activities that families will love. The resource has also been translated into Chinese and Punjabi.

I am pleased to announce the release of another updated resource — Mastering the Facts: Addition. In this 2nd edition you’ll find new games, updated strategy lessons and more literature links to support students in learning — and retaining — their addition facts.

There is no doubt that students need to master their multiplication facts — and how we get them there matters.

In this revamped second edition of Mastering the Facts Multiplication, you’ll find a scope and sequence for teaching the facts, strategies for addressing each of the fact families, expanded whole group lessons, new games for practice and literature connections to ensure your students learn — and retain — the facts.

I would like to acknowledge and recognize those teachers from SD72 Campbell River who helped to develop these lessons for the original volume in 2012. Our collaboration continues to inspire!

For those who have been patiently waiting (and lovingly nagging!) I am happy to announce the publication of my newest math teaching resource: Proportional Reasoning in Intermediate for Grades 4-8, available now through my online store.

This resource for teachers of Grades 4 to 8 presents more than 250 pages of open-ended lessons, meaningful practice, games, literature connections and a wealth of problem-solving contexts for supporting students to make sense of fractions, decimals, percentages, ratios and proportions. Designed for today’s diverse classrooms, this resource offers a range of tasks to promote proportional thinking through intentional development of mathematical language and the use of key manipulatives. Colour tiles, Cuisenaire rods and tangrams are used to model and make connections within and between concepts.

In the first section of the resource, students will explore three models – set, area and linear – for representing and describing, comparing and ordering fractions. Students will learn to convert between fractions, decimals and percent and to apply these skills to problem situations, including measurement, tax, discounts and data management. Next, students learn to add and subtract fractions, to solve proportions using a range of strategies (involving both mental math and the appropriate use of technology) and then finally to multiply and divide fractions.

Assessment tools are threaded throughout the resource to allow teachers to keep track of student progress and to make instructional decisions.

Proportional reasoning IS the math we do every day. This resource provides an access point for all.

In these new French translations of the English originals, you’ll find more than 200 open-ended and engaging problems for french immersion students from primary through middle school. All are posed in French and explore important mathematical concepts across the grades.

Like its intermediate counterpart, this compact but potent book comes with an easel so you can set it up on your desk and flip from one rich problem to the next, posing open-ended questions of your primary students.

Good Questions: A Year of Open-Ended Math Problems for Grades 2-4 is a problem-a-day resource that includes rich tasks ideal for grades 2, 3 and 4. Organized by topic and structured in problem sets of 5, this simple to use teacher resource includes 200 mathematically important questions to engage your students in deep thinking. For only $25, it’s a reasonably priced way to stimulate and promote mathematical conversation!

Operations, measurement, proportional thinking and patterns are featured in this calendar of problems. Each one engages students in thinking flexibly, critically and creatively to solve tasks of varying complexity.

This compact but potent book comes with an easel so you can set it up on your desk and flip from one rich problem to the next, posing open-ended questions of your intermediate students.

Good Questions: A Year of Open-Ended Tasks is a problem-a-day resource that includes
rich tasks ideal for grades 5, 6, 7 and 8. Organized by topic and structured in problem sets of 5 or more, this simple to use teacher resource includes 210 mathematically important questions to engage your students in deep thinking. For only $25, it’s a perfect back-to-school gift for yourself!

Proportional reasoning, measurement, operations and algebra are featured in this calendar of problems. Each one engages students in thinking flexibly, critically and creatively in the face of important and challenging mathematics.

My sincere appreciation to all of you who have waited for the publication of this book. As you know, I’ve had a pretty remarkable year. I hope you’ll forgive me, knowing that only good distractions delayed its completion!

But I am pleased as punch to announce the release of Place Value For Intermediate: Building Number Sense for Grades 3-5, available now from my online store for $50.

This resource for teachers of Grades 3 through 5 features lessons designed to support deep learning of number. A wide range of both open-ended and directed tasks focus on representing, describing, comparing and ordering numbers to 100 000, as well as explorations of decimal numbers to thousandths.

Measurement experiences make up a big part of this series of tasks. The metric system and all of its place value connections is featured in explorations of linear measurement (mm, cm, m, km), perimeter (cm, mm), area (square cm and square m), mass (g, kg) and capacity (mL, L).

Addition and subtraction of large numbers and decimals are also addressed in this volume. Lessons at the grades 4 and 5 level focus on multiplication of 1 by 2- and 3-digit factors as well as 2 by 2-digit factors using the distributive property (an area model).

Assessment tasks tap into students’ understandings of these numbers and their application in the real world. Being able to see and relate to big numbers and to very small ones, to understand their relative size and to capably use these numbers to estimate is the essence of number sense.

Set up in a developmental continuum intended to facilitate the teaching of combined grades, this 352 page volume is certain to contain material to meet the needs of all learners and to inspire fun and engagement with critically important place value concepts.

When you buy the book online, you also get access to almost 40 pages of digital files and resources, which will be emailed to you as a downloadable pdf!

I am truly grateful for the support, encouragement and professional curiosity of those of who who have visited my blog since its inception so many years ago… I am indebted to those of you who have told friends and colleagues about the blog and grateful for my “regulars” — those of you who return to it from time to time, looking for resources. Your enthusiasm buoys me up!

It is indeed a great privilege to be a part of this international community of teachers of mathematics. Thank you.

Wow. I have had the most extraordinary summer. Truly extraordinary. And somehow between engaging in a series of remarkable, life-affirming adventures I have managed to write another teacher resource book… 😊

It’s all about Place Value (as I’m sure you’ve figured out!) and is intended for teachers of kindergarten through grade 2, with special accommodations for those who teach in combined grades settings. There are 230 pages of developmentally framed lessons designed to address the diversity in our primary classrooms. Each one supports students to represent and describe quantity, to compare and order sets, to use referents to estimate and to skip count. Lessons devoted to measurement — an ideal practical application of place value in the world — are also featured. Whole class lessons, centres tasks and games for practice allow students to connect these important concepts in a seamless way, and can be used both as a unit or spread throughout the year to build and consolidate understanding.

I thought it was time to post another game for those of you who are looking to support your intermediate students. This is another classic game from BEAM. It’s called the Game of Remainders — but don’t be fooled! It’s about far more than simple division. There are connections to be made to skip counting and the multiples here that are worth talking about!

As a tool for thinking and for identifying the important patterns inherent in this game, consider giving students a hundred chart to begin. Have them shade or highlight all the multiples of 6 (6, 12, 18, 24, 30, 36, etc) before playing the game.

Then, as they land on a number in the wheel (like say 49), they can refer to the chart and see that the number 49 is not coloured, so it’s going to have a remainder. Looking further, the will notice that it is in fact one more than a multiple of 6, which means there will be 1 remainder.

Fun, right?

Be sure students gave a chance to talk about what they’re noticing in the chart as they use it. The more we describe our thinking, the clearer it gets and the more connections we make!

I’ve made a few other versions of this game if you’re interested in downloading them. They follow the same format, but address divisibly of 3, 4 and 5.

There are twelve days of Christmas. And according to the song, there were a lot of gifts given over those 12 days.

Which means there’s a math problem lurking in there!

Have students think about ways to calculate the total number of gifts given on the 12th day (12+11+10+9…) or even calculate the total number of gifts given throughout the entire gift-giving season (the gifts from the 12th day added to all the gifts given on the 11th day…). Some may even apply ideas borrowed from Gauss to effortlessly calculate the sum of the arithmetic sequence.

And then if you want to get especially sassy, challenge students to calculate the daily cost of giving each of these extravagant gifts! Tamara B. shared this awesome website with prices for the items given over the 12 days – including shipping and handling changes – so that you too can figure out what it would cost to give 5 gold rings, 4 calling birds, 3 french hens, 2 turtledoves and a partridge in a pear tree… not to mention financing the maids a-milkin’!

The site is only updated to 2003, but at least it will provide a starting point. But better yet… If inflation has increased on average 2% a year for the last 12 years… well, the mathematical possibilities are endless! Consider checking out the Statistic Canada webpage for the actual inflationary numbers. And hold onto your calculators!

Tonight marks the first evening of Hanukkah – the Festival of Lights in the Jewish tradition. Between candle-lighting and spinning tops, there is a great deal of math in this 8-day festival! Consider the following seasonal activities for your students in the days ahead. I hope you enjoy them!

There are 8 days of Hanukkah. For each of the 8 days, a candle is lit and placed in the menorah – one on the first day, 2 on the second day etc. Sounds simple, right? Well, yes and no. By the end of each day, these candles burn out and have to be replaced, which means that there are many candles used throughout the 8 days of Hanukkah. To top it all off, each of these candles is lit by another candle, called the shamash, which is then placed in the centre of the menorah.

So. If you had to buy enough candles for this year’s Hanukkah celebration, how many would you buy in all?

Young children might enjoy using materials to model and solve the problem. Older students might use some Gauss-ian logic and/or ideas around triangular numbers to generate a solution…

Playing the dreidel game is another fun – and mathy – way to celebrate Hanukkah. If you’re looking for the complete rules and a bit of history of the game, click here for more information.

In essence, this is a gambling game that involves a set of prizes. In the classroom, consider using counters, cubes or beads as “ante”. Players spin the 4-sided dreidel top and follow the instructions listed on each side, taking all, none or half of the pot – or adding counters to keep the game going! Even young children will enjoy playing this game and practicing concepts of addition, subtraction and “half of”.

I wanted to share some fun puzzle templates with you — one set for primary classrooms and another for intermediate. Both require the same reasoning skills: students must use a complete set of 0 to 9 tiles and place them on each of the blank spaces provided. Some of the forms are more complex than others, and will require students to not only know their facts but also to reason through the logic of placing the tiles in the correct position.

While students work, wander the classroom and ask them how they are making decisions about the placement of the number tiles. Students’ rationale may surprise you!

Some of my colleagues have suggested different ways to make the tiles:

by cutting up vinyl placements into 1′ by 1″ squares and using Sharpie to record the numerals

For those of you about to return to another school year, welcome back!

I am truly excited to announce the release of my newest teacher resource book: Multiplicative Thinking: From Skip Counting to Algebra (Grades 3 to 8). This book is designed for teachers of the intermediate grades and is focused on the teaching and learning of multiplication. This resource addresses multiplication deeply — what it means to multiply, when to use multiplication in problem-solving situations, as well as how to manipulate whole number, fractional and decimal factors using strategies like the distributive property.

Lessons on skip counting, patterns in the multiples, factoring, and on prime and composite numbers are included in this 220 page teacher resource. Algebraic thinking is explored as well, from T-charts and input-output machines to solving equations, from graphing linear relations and extrapolation to finding the slope of a line. Students engage with visuals and real-world problems involving proportionality, rates, discounts and taxes to build their understanding of multiplicative thinking and see its very real application to their everyday lives.

Each of the 40 lessons features a connection to prior knowledge, whole class and small group explorations of the Big Math Ideas, guided conversations about the mathematics with key vocabulary, opportunities for meaningful practice, tasks for consolidation and customized assessment tools. Skill building lessons are interspersed throughout the book, ensuring students recall and continue to practice the essential skills needed to apply multiplicative ideas.

And of course literature links and games for practice are — as always — included!

Multiplicative Thinking: From Skip Counting to Algebra (Grades 3 to 8) is available for $40 + $10 expedited shipping. To order, click here or on the link at the right. From there you can also order other titles, including Mastering the Facts: Multiplication, a resource dedicated to the teaching and mastery of the critically important multiplication facts. It’s a perfect complement to this new volume and one that can be used in advance — or concurrently — to build a solid foundation.

Thank you for your support. All the best for a remarkable school year!

Carole

Why Multiplicative Thinking?

Multiplicative thinking plays an enormous role in elementary and middle school mathematics. So much bigger than simply knowing the facts — a critically important aspect — the ability to think multiplicatively is essential for success with almost every other mathematical concept, from ratio and proportionality to algebra. It is the operation most often used in “real life” to make sense of large quantities, of taxes and discounts, of income per hour and kilometres travelled. It’s the operation we use when we figure out how much paint or carpet to buy or what a tank of gas is going to cost; when we convert currency for a holiday away or sort out how much to tip on a meal. No matter where we look, multiplicative situations abound. We can’t spend too much time on the teaching and learning of these critical concepts!

In writing this resource, I have attempted to introduce multiplicative thinking — both the operation itself and the bigger concept of multiplicative reasoning — in a sense-making way. Through stories, models, pictures and words, students are introduced to the idea of multiplication as “groups of” and as “rows of”. Problems are posed to support learners in connecting what they know about patterns in the multiples to proportional situations. The associative and distributive properties are introduced and applied. Algebraic concepts — input and output machines, graphing and exploring the rate of change in linear relations — round out the topic and provide a preview for multiplicative reasoning at the middle and high-school levels.

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Welcome!

I am Carole Fullerton, an independent consultant working with teachers around British Columbia (and beyond!) in the area of numeracy. I work with districts, whole school staffs, with school-based learning teams, in classrooms and with parents in an effort to promote mathematical thinking.